4. THEORY

The detailed theoretical understanding of Type Ia Supernovae is still
limited. Two very complicated physical processes are at work in SNe Ia
explosions. First there is the explosion mechanism itself, which is
still debated and several possibilities are proposed and then there is
the complicated, highly non-thermal process of the radiation escape which
leads to the observed phenomenon. A recent review of SN Ia theory is
presented by
Hillebrandt &
Niemeyer (2000).

In general it is agreed that SNe Ia are the result of thermonuclear
explosions in compact stars. White dwarfs are favored by their intrinsic
instability at the Chandrasekhar mass and the fuel they provide in
carbon and oxygen. All the arguments for this scenario have been already
clearly laid out before 1986
(Woosley & Weaver
1986
and references therein). Other fuels could be imagined, but all of them have
some problems. They either do not provide enough (explosive) energy
(like hydrogen)
or can not synthesize the intermediate-mass elements (like helium, which
detonates). Higher elements are in principle possible, but it is well
known that O-Ne-Mg white dwarfs would rather collapse to a neutron star
than explode because of the large electron capture effects (e.g.
Nomoto & Kondo
1991,
Gutiérrez et
al. 1996).
The initiation of the burning in the degenerate star is, however, a puzzle.
For many years it was clear that a detonation (supersonic burning front)
would lead to an overabundance of iron-group elements and not enough of the
intermediate-mass elements observed in the spectral evolution during the
peak phase. A deflagration (subsonic burning) seemed more appropriate,
but it was not clear how to prevent the explosions to turn into a
detonation. The phenomenological model W7
(Nomoto et al. 1984,
Thielemann et
al. 1986)
or similar explosions
(Woosley & Weaver
1986,
1994b)
enjoy a great popularity as the explosive input model for spectral
calculations since they seemed to reproduce the element distribution fairly
accurately (e.g.
Harkness 1991,
Jeffery et al. 1992,
Mazzali et al. 1993,
1995,
1997,
Yamaoka et al. 1992,
Shigeyama et
al. 1994).
The burning speed in this model has, however, never been understood
in physical terms.
Possible alternatives are the pre-expansion of the white dwarf to lift
the degeneracy by a slow deflagration first and have the detonation start
later
(Khokhlov 1991).
The critical parameters in these models are the
density at the transition from deflagration to detonation,
the pre-explosion density, the chemical
composition (mostly C/O ratio), and the deflagration speed at the
beginning of the burning. The transition density has been proposed as
the critical parameter for the nucleosynthesis and hence the amount of Ni
produced in the explosion.
These delayed-detonation models can reproduce some of the observations
(Höflich 1995,
Höflich &
Khokhlov 1996,
Höflich et
al. 1996).
However, their consistency has been questioned recently
(Niemeyer 1999,
Lisewski et
al. 1999a,
b).
Another
possibility is that the first explosion in the center fizzles and as
the star contracts again, the density and temperatures rise high enough
to re-ignite carbon near the center and lead to the explosion
(Arnett & Livne
1994a,
b,
Höflich et
al. 1995).
There are hence several
theoretical possibilities to ignite the white dwarf, but it is still not
clear which ones are realized in nature. With the variety of
SN Ia events observed now, it is possible that SNe Ia come from
different burning processes. However, the observed correlations must
then be valid across different explosion mechanisms.

Once the explosion has started, the flame has to continue burning enough
material to unbind the star. In many calculations this has not occurred
and the flame has fizzled. Only recently have some three-dimensional
calculations led to weak explosions
(Khokhlov 1995,
Niemeyer et al. 1996,
Reineke et
al. 1999).

An altogether different explosion mechanism on sub-Chandrasekhar mass white
dwarfs has been explored
(Nomoto 1982,
Livne 1990,
Livne & Glasner
1991,
Woosley & Weaver
1994a,
Livne & Arnett
1995).
In this model, the explosion is generated at the surface
of the white dwarf due to a detonation of He at the bottom of the
accretion layer. This model solved the progenitor problem by allowing
explosions well below the Chandrasekhar mass near the peak of the white
dwarf mass distribution. Difficulties here are the initiation of the
explosion and the subsequent ignition of the whole star by a pressure
wave. Many of these calculations are still parametric and the details
have to be worked out (cf.
Woosley 1997).

Another complicated process stands between the explosion models and the
observations. The release of the photons from the explosion is
computationally extremely difficult to follow. The reasons are the
continuous change of the energy deposition and the detailed physics of the
conversion of the
-rays
injected inside the ejecta from the
radioactive decays to the low-energy photons observed. The opacity
changes due to the thinning of the expanding ejecta for the high-energy
input, but at the same time the high velocities and the abundance of
higher elements with their large number of transitions complicates the
calculations
(Harkness 1991,
Höflich et
al. 1993,
Eastman 1997,
Pinto & Eastman
2000).
The exact treatment is still debated, but it has become increasingly clear
that the old assumption of a thermal input spectrum is not tenable. Even
though SNe Ia display a nearly thermal 'continuum' during their peak
phase, they are really dominated by the time-dependent photon
distribution. The clearest demonstrations of this fact are the lack of
photons in the J-band
(Spyromilio et
al. 1994,
Meikle 2000)
which is
due to the absence of emission lines in this wavelength region and the
occurrence of the maximum in different optical filters, which is
reversed for most supernovae, i.e. the near-IR filter curves peak before
the optical ones
(Contardo et al. 2000,
Hernandez et
al. 2000).

Due to the large opacities in the ejecta the photon degradation proceeds
through several channels (e.g.
Lucy 1999,
Pinto & Eastman
2000).
Since the UV region is blocked by many velocity-broadened iron-group lines
(Harkness 1991,
Kirshner et al. 1993),
the photons are progressively redshifted
until the optical depth is small enough for them to escape. This occurs
first in the near-IR and hence the peak is reached earlier at these
wavelengths
(Meikle 2000,
Contardo et al. 2000).
However, only in wavelength regions where plenty of line transitions in
the outer layers are available is there any significant flux.

It will take a few more years until these problems can be addressed
completely. A closer link between the observations and the models has
been pursued by trying to understand the correlations which have been
observed. The light curve decline has been modeled
(Höflich et
al. 1996)
and explained as due to differences in the amount of Ni produced in the
explosion. Also the color dependence could possibly be explained
this way. Other issues like the rise time or the occurrence of the
secondary peak in the near-IR remain, however, open. A possible
interpretation of the light curve stretching during the peak phase and
for the bolometric light curves links the time scales of the Ni decay,
the diffusion time (for a constant opacity) and the age of the supernova
(Arnett 1982,
Arnett 1999).

By comparing the kinetic energy as derived from line widths and the
measured Ni masses it should be possible to derive global parameters of
the explosion. First such steps have been made by
Mazzali et
al. (1998),
Cappellaro et
al. (1998), and
Contardo et
al. (2000).
This alternative route will not replace the
detailed modeling of light curves and spectra, but may provide a more
direct input for the explosion models.